- Email: [email protected]
The fluid dynamics of Balanus glandula barnacles: Adaptations to sheltered and exposed habitats
Journal of Biomechanics 71 (2018) 225–235
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
The fluid dynamics of Balanus glandula barnacles: Adaptations to sheltered and exposed habitats Maureen Vo a, Sasan Mehrabian a, Fernando Villalpando b, Stephane Etienne b, Dominique Pelletier b, Christopher B. Cameron a,⇑ a b
Departement de sciences biologiques, Universite de Montreal, Montreal, Quebec H3C 3J7, Canada Ecole Polytechnique de Montreal, C.P. 6079, succ. Centre-ville, Montreal, Quebec H3C 3A7, Canada
a r t i c l e
i n f o
Article history: Accepted 6 February 2018
Keywords: Balanus glandula Barnacles Leakiness Feeding Computational fluid dynamics
a b s t r a c t Suspension feeders use a wide range of appendages to capture particles from the surrounding fluid. Their functioning, either as a paddle or a sieve, depends on the leakiness, or amount of fluid that passes through the gaps between the appendages. Balanus glandula is the most common species of barnacle distributed along the Pacific coast of North America. It shows a strong phenotypic response to water flow velocity. Individuals from exposed, high flow sites have short and robust cirral filters, whereas those from sheltered, low velocity sites have long, spindly appendages. Computational fluid dynamics (CFD) simulations of these two ecophenotypes were done using a finite volume method. Leakiness was determined by simulating flow velocity fields at increasing Reynolds numbers, results that have been unattainable at higher velocities by observation. CFD also allowed us to characterize flow in hard to see regions of the feeding legs (rami). Laser-illumination experiments were performed at low to medium flow velocities in a flume tank and corroborated results from CFD. Barnacle filters from a sheltered site become completely leaky at Re ¼ 2:24 ð0:16 m=sÞ, well above the maximum habitat velocity, suggesting that this ecophenotype is not mechanically optimized for feeding. Barnacles from exposed environments become fully leaky within the range of habitat velocities Re ¼ 3:50 ð0:18 m=sÞ. Our CFD results revealed that the drag force on exposed barnacles feeding appendages are the same as the sheltered barnacles feeding appendages despite their shape difference and spacing ratio. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Suspension feeders use a diverse range of filamentous appendages to capture particles from the surrounding water (Mehrabian et al., in press; Riisgård and Larsen, 2010). Most appendages essentially consist of arrays of bristles or cylinders, and some behave as sieves to intercept particles from suspension passing through (e.g. bands of cilia, arthropod setae, and mucous nets), while others create feeding currents or act as paddles to direct pockets of fluid for further processing (e.g. flagella, cirri, or tentacles) (Riisgård and Larsen, 2010; Vogel, 2003). The performance of these appendages (i.e. paddle vs. sieve) depends on their interactions with the surrounding medium. Fluid can either pass readily through the gaps between filaments, creating a ‘leaky’ appendage that functions as a sieve, or it may flow around the perimeter of the appendage that functions as a paddle.
⇑ Corresponding author. E-mail address: [email protected] (C.B. Cameron). https://doi.org/10.1016/j.jbiomech.2018.02.011 0021-9290/Ó 2018 Elsevier Ltd. All rights reserved.
The feeding behaviour and flow about several species of barnacles have been described, especially in Balanus glandula (Geierman and Emlet, 2009; Marchinko, 2007; Marchinko and Palmer, 2003; Miller, 2007; Southward, 1955; Trager et al., 1990). B. glandula (Fig. 1) is a common Northeast Pacific species of acorn barnacle found in a wide range of wave-exposed habitats. It has been subject to questions about how morphology and fluid dynamics affect appendage functioning. Its feeding legs demonstrate a great degree of phenotypic plasticity in response to water velocity (Marchinko, 2007; Marchinko and Palmer, 2003; Southward, 1955). Its sessile nature facilitates easy observations and manipulation in the field or lab (Arsenault et al., 2001). The parameters of the feeding structures, including rami length and width, and seta length and spacing, of individuals found at low and high flow velocities is well characterized (Arsenault et al., 2001; Marchinko and Palmer, 2003). Less well known are how the water and appendage interact in confined, small, and hard to see regions close to the appendages. Also, observations at realistic, high flow velocities have not been
226
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
Fig. 1. Microscopic image of the entire feeding appendage of A: Exposed and B: Sheltered Balanus glandula. I, II, III, IV, V, and VI refer to the cirrus numbers. Each cirrus consists of 2 rami. The box shows the portion of cirri VI used for CFD modeling. C: Microscopic image of a ramus and segments of a feeding appendage. D: Frontal view of the two rami of cirrus VI. E: Schematic front view of a ramus segment. h is the length of the segment, w is segment width, b is the setae diameter, i is the centre-to-centre setae spacing, s is the setae length, and l is segment length, which is in the flow direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
reported in the literature. Several mathematical and physical models of fluid velocity profiles around rows of cylinders of finite width have demonstrated that the transition to leakiness depends largely on the Reynolds number and setae spacing ratio (Cheer and Koehl, 1987a,b; Hansen and Tiselius, 1992; Koehl, 1993; Loudon et al., 1994; Mehrabian et al., in press; Tamada and Fujikawa, 1957). The Reynolds number represents the ratio of inertial to viscous forces:
Re ¼ Ub=m
ð1Þ
where U is the free flow velocity, b the characteristic length, in our case the base diameter of the longest setae (see Fig. 1E) and m the kinematic viscosity of the surrounding fluid. For an incompressible isothermal flow, this number fully characterizes the fluid flow around a given geometry pattern, like a filament (Vogel, 2003). For most biological filaments, the Reynolds number ranges from
105 to 10 (Cheer and Koehl, 1987b; Jorgensen, 1983; LaBarbera, 1984; Rubenstein and Koehl, 1977; Shimeta and Jumars, 1991). At low values, viscous forces dominate, velocity boundary layers are thick relative to the dimensions of the body, and the flow about the appendage is laminar (Happel and Brenner, 2012; Vogel and Savage, 1996). But, from dimensional analysis, when geometry varies, geometrical ratios have to be considered. For example, the setae spacing ratio, i=b (the ratio of setae center-to-center spacing, i, to setae diameter b, see Fig. 1E) increases from sheltered to exposed barnacles and is known to have an important effect on leakiness (Marchinko and Palmer, 2003). In general, for Re ¼ 0:5, high leakiness is observed when i=b becomes larger than 5 (Cheer and Koehl, 1987b; Koehl, 1996, 2001). However, for Re of the order of 103 , changes to this ratio has little influence on leakiness (i.e. morphological and behavioural diversity without performance consequences) (Koehl, 1996).
227
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
The mechanisms and rates of particle capture by filter feeding appendages depend on the flow field around the individual filaments of a filter (Rubenstein and Koehl, 1977; Shimeta and Jumars, 1991). Specific morphological changes and novel functions can result simply from changes in size, shape, stiffness, or habitat of an organism. To understand the relationships between morphology and functioning and the basic physical rules governing how a biological structure operates, we embarked on a mechanical study. The objectives of this study are to compare the flow dynamics, forces (drag) on the barnacle cirri, and the leakiness, between passive feeding barnacles from exposed (i.e. robust phenotype) and sheltered (i.e. spindly phenotype) shores across a range of realistic flow velocities. In this paper, we study the appendage-water interaction for the robust, and spindly barnacle cirri phenotypes. We visualize these poorly known interactions, and characterize them using dimensionless numbers, in low-velocity, medium velocity, and high velocity scenarios for the spindly phenotype, and the robust phenotype. We accomplish these goals using a computational fluid dynamics (CFD) tool, and validate our results by comparing our findings to observations of B. glandula exposed to flow at different velocities experimentally in a flume tank. The information gained from this comparative approaches (CFD model, and observations) is significant because it precisely quantifies the amount of force applied to the appendages and the amount of water that flows between the appendages (leakiness) at different flow velocities. Also, we are able to visualize the flow behavior around B. glandula rami across a range of realistic velocities and determine the velocities at which each phenotype transitions from paddle to ‘leaky’ sieve. 2. Experimental materials and methods 2.1. Collection sites Balanus glandula specimens were collected from two sites in Barkley Sound, Vancouver Island, British Columbia, Canada
between May and July, 2013 (Table 1). Rocks and mussels (Mytilus californianus) with attached barnacles were collected from the sheltered and low flow shoreline of Grappler Inlet, and from the wave exposed and high flow shoreline of Seppings Island (Table 1). Careful attention was paid to choose barnacles from microniches that were not obviously obstructed by adjacent obstacles. These collection locations are the same as those used by Marchinko and Palmer (2003). Table 1 reflects average maximum velocity rates recorded by Marchinko and Palmer (2003). Barnacles were transported to the Bamfield Marine Sciences Centre and maintained in sea water Table (181 cm long 84 cm wide 8 cm deep). Barnacle specimens used for fixing, measuring and testing were used immediately upon collection or within the first two days. 2.2. Fixing of barnacles Barnacle specimens were removed from their shells and placed in a 7% solution of MgCl2 to relax the feeding legs (rami). Specimens were then transferred to Bouin’s fixative for 24 h to fix legs in the open, passive feeding position and stored in 70% ethanol. 2.3. Cirral anatomy measurements As phenotype varies widely in B. glandula, the goal was to characterize the two drastically different shapes for flow observations and CFD analysis that represent sheltered and exposed barnacles. Six barnacle samples from exposed environments, and six barnacles from sheltered environments, were used to obtain averages for the parameters used to construct the CFD model (see Table 2). The posterior three cirri (IV, V, VI) were selected since they are the longest and primary food capturing appendages (Marchinko and Palmer, 2003). Individual barnacles were observed with an Olympus Q Color 5 camera mounted on an Olympus SZX16 dissecting microscope. Images, similar to those shown in Fig. 1, were captured using Q capture software (Surrey, BC, Canada). The cirrus with the longest rami, cirrus VI, was removed, straightened into the feeding position, and the ventral surfaces and side profiles were
Table 1 Collection sites and descriptions. Wave exposure
Site
Sheltered Exposed ⁄ ⁄⁄
Grappler narrows Seppings Island
Location description
North latitude⁄⁄ 0
West-facing shores Western-most rocks on southwest facing
00
48°49 91 48°500 5000
West latitude⁄⁄ 0
Water velocity⁄ (m/s)
00
125°07 64 120°120 5000
0.0052 4.41
Adapted from Marchinko and Palmer (2003). Barnacles collected from Barkley Sound, Vancouver Island, British Columbia, Canada.
Table 2 Summary of sheltered and exposed barnacle dimensions. Each data point is an average of n ¼ 6 barnacle samples. Barnacle type
Setae
Segment
Setae number⁄
Length s (lm)
Base diameter b (lm)
Tip diameter t (lm)
Spacing ratio i=b
Angle h (deg)
Length l (lm)
Height h (lm)
Width w (lm)
Sheltered (n = 6)
1 2 3 4 5 6
515 58 363 39 224 21 147 17 90 11 62 12
15 4 16 3 13 2 11 2 81 51
4 0:78 4 0:78 2 0:50 2 0:50 2 0:50 0:5 0:20
1.50 1.69 1.71 1.82 2.29 3.00
47 2
186 13
216 19:1
93 19
Exposed (n = 6)
1 2 3 4 5 6
445 14 314 87 275 68 161 64 106 28 65 20
20 2 20 2 18 1 15 2 10 2 6 1:5
5 1:30 5 0:82 4 0:78 3 1:30 3 0:79 2 0:50
1.20 1.08 1.09 1.25 1.45 2.50
51 3
280 44
164 24
98 8
is the standard deviation of the averaged data points. ⁄ Each setae number refers to an individual setae as illustrated in Fig. 1E.
228
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
photographed (Fig. 1A–D). Morphometric analysis using ImageJ (developed by the National Institutes of Health, Bethesda, MD, USA) of enhanced micrographs and scanning electron microscope images of central segments of cirri VI produced values for segment length, width, and height, setae lengths and widths (at the proximal and distal ends), angle of setae with respect to the cirri segments, intersetal spaces, and distance between cirri (Fig. 1A–C, Table 2).
2.4. Laser-illumination flow visualization Flow visualization was performed in a laboratory flume having 107 cm length, 18 cm width and 19 cm depth (see Supporting Fig. 1), at a water temperature of 12 °C. Extracted barnacle bodies were held 70 cm from the start of the chamber with tweezers embedded in modelling clay, such that the feeding legs were free to open in the passive feeding position, in a mid-water position
Fig. 2. GAMBIT models illustrate front and top view of sheltered and exposed barnacles. Computational model of six cirral segment and setae of B. glandula with boxed domain boundary, showing the direction of flow (green arrows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
of the flume. A class IIIb, wavelength 532 nm laser (Laserglow technologies) was passed through a concave crystal to create a two dimensional horizontal sheet across the mid-section of the barnacles cirri for flow field observations. The flume water was seeded with silver-coated neutrally buoyant hollow glass particles of 11 lm in diameter (110P8, LaVision GmbH) at a concentration of 10:50 g=ml. The particles were neutrally buoyant, and therefore motion was only horizontal, in the direction of flow. Fully developed flow fields around filters were recorded using a Sony HDV (1080i HVR-Z1U/Z1N, 60i) video recorder at a rate of 30 frames per second. Water velocity was determined by frame-byframe video analysis of suspended particles across a distance of 20 cm, taken 10 cm before the passively positioned barnacles. Only a few particles, that could be reliably tracked, were used to establish flow speeds, which were constant. At times, particle visualization was improved by injecting a high density of particles directly upstream of the feeding legs using a pipette. A variable speed motor (90 volt, 7.6 amp) and propeller was used to control flow velocity that ranged from 0.03 m/s to 0.19 m/s. Leakiness was determined once particles were observed to pass directly through the barnacle appendages. Like those who studied barnacle appendage-water interaction systems before us, we were unable to see flow in confined spaces, nor at high velocities. Also at Reynolds numbers higher than 6, particles motions were too fast to evaluate velocity vectors accurately. iMovie ’11 (version 9.0.4) and ImageJ (Schneider et al., 2012) were used to analyze the video footage. 3. CFD model and fluid flow simulations In this section, we carried out numerical simulations of the flow around outer rami, which are exposed to free flow, and between opposing rami, where setae interdigitate, for the two phenotypes (sheltered and exposed). Our objectives were to (1) visualize flow pathlines, and velocity vectors/contours in confined spaces of the barnacle appendages, (2) to obtain the drag force applied to the appendages at various Re, and (3) to simulate realistic flow conditions to determine the Re at which the appendages transition from paddle to sieve. The accuracy of our simulations were verified against our laser-illumination visualizations. 3.1. Three-dimensional models Each cirrus has two rami (legs). Six segments of one and a half rami were modeled as elliptical cylinders (Fig. 2). The segments vary in height, width, and length characteristic of each barnacle phenotype (Table 2). All geometries were converted to dimensionless form, to facilitate scale-up of obtained results to real conditions and to simplify and generalize the numerical calculations. The base diameter of the longest setae from the sheltered barnacle was used as the reference number to divide all parameters of both models. This converted all our models from dimensional to dimensionless scale for simulations. Projecting anterior-laterally from each segment were six pairs of conical-shaped setae, which protruded from the segments at an angle of 47 for the sheltered model, and 51 for the exposed one, with respect to the corresponding pair (Fig. 1D). Distances of 500 lm between sheltered rami, and 400 lm between exposed rami were chosen for the CFD model. These distances are realistic, and represent the smaller spacing of the exposed phenotype. Specific segment parameters are summarized in Table 2. As illustrated in Fig. 1, the distal two setae interdigitate with those of the adjacent rami, whereas a gap was modeled between the shorter, more basal, setae. In order to minimize memory requirements and computational time, six segments are modeled, and one of the rami (elliptical cylinders)
229
was divided in half and reflected as a mirror image using the ’symmetrical’ function when defining boundary conditions. The other segment was left whole to examine flow behavior between the interdigitating setae, and around the outer setae that was exposed to a free flow stream (Fig. 2). 3.2. Meshing The 3D spatial discretization of the two barnacle filter models were generated using GAMBIT (Fluent 13, ANSYS, 2012, Cecil Township, Pennsylvania, USA) a Fluent CFD pre-processing program. A variety of verification tests were performed to assess the appropriate number of cells and the fluid domain size to use in the CFD simulations. A width of 80 diameters of the reference model, which is the base diameter of the longest setae from the sheltered barnacle, was found to have negligible blockage effect. Three mesh sizes of approximately 3, 7 and 9 million cells, standing for coarse, medium, and fine meshes respectively were tested on each of the barnacle models. The mass flow rate at the segment was monitored for each case and a final mesh size at which increasing the grid size showed no significant changes in mass flow rates was found to be 6,808,487 cells for the sheltered barnacle model and 7,439,983 cells for the exposed one. More specifically, each setae tip and base is defined with 20 nodes around the circumference and nodes along the length of setae ranging from 250 to 45 from longest to shortest setae, respectively. All setae and the region directly surrounding setae are areas of high interest to understand the feeding mechanics of B. glandula. Therefore, small cells were inserted in this region. Boxed areas were created around the filter structures to control the cell size in critical areas (Fig. 2). A size function using setae as the source of growth allowed cell size to increase farther away from the source at a growth rate of 1.05. 3.3. Boundary conditions The steady-state incompressible Navier-Stokes equations which represent conservation of mass and momentum, can be solved provided that boundary conditions are prescribed on all surfaces surrounding the fluid domain. The boundary conditions imposed in these simulation were as follows: Inlet conditions: constant horizontal speed (uniform velocity profile). Outlet conditions: pressure outlets. Left wall: symmetry conditions. Right wall: wall boundary condition (no-slip boundary condition). Barnacle/water interface: wall boundary condition (no-slip boundary condition). The fluid is seawater, and was considered as incompressible with a density of 1024 kg/m3 and a kinematic viscosity m of 1:047 106 m2 =s at 20 °C (Vogel and Savage, 1996). 3.4. CFD model The commercial CFD software package ANSYS FLUENT (version 13.0) based on a finite volume method was used to solve the threedimensional Navier-Stokes equations. In this method, the field of calculation is divided into a certain number of meshes, called control volumes. The system of equations governing the flow (continuity and momentum equations) is discretized at the level of these meshes, which leads to a system of non-linear algebraic equations solved in an iterative way.
230
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
Our simulations were carried out for three flow velocities (U ¼ 0:04; 0:21 and 0:50 m=s). These inlet velocities correspond to the flow velocity range used in previous B. glandula feeding studies (Marchinko, 2007; Marchinko and Geary, 2003), and measured in situ (Geierman and Emlet, 2009; Trager et al., 1990). Also, the velocities tested in the laser illumination experiments are within
the lower end of this range (i.e. U ¼ 0:03—0:19 m=s). Furthermore, given the small size of the barnacle filters (see Table 2) and the flow velocities considered, the Reynolds number defined in Eq. (1) is of the order of 101 to 100 . This implies that the flow around the barnacle filters is laminar. As a consequence, the SIMPLE algorithm, QUICK momentum scheme, ‘Green-Gauss cell
Fig. 3. Top view of velocity contours and magnified velocity vectors passing through intersecting setae with the direction of flow from right to left for sheltered barnacles. Green arrows indicate flow direction. Velocities in the scale bar are normalized with respect to the free stream velocity U. To calculate the real velocities multiply these numbers by the inlet velocity.
Fig. 4. Top view of velocity contours and magnified velocity vectors passing through intersecting setae with the direction of flow from right to left for exposed barnacles. Green arrows indicate flow direction. Velocities in the scale bar are normalized with respect to the free stream velocity U. To calculate the real velocities multiply these numbers by the inlet velocity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
231
Fig. 5. Laser-illumination of water flow and wake behavior viewed from above a passive B. glandula feeding appendage. The direction of flow is from right to left. A–C is a sheltered barnacle and D–F is exposed.
based’ gradient and pressure-velocity coupling method were used to solve the problem. The sum of normalized residuals of all variables were considered converged when lower than 104 . This occurred within 900–3000 iterations for all models tested. Our CFD results were quantified based on two parameters: drag force F D and leakiness. Drag force is the sum of the stresses and pressure over the surface area of the feeding appendage. In this study, the drag coefficient C D was obtained from FLUENT to calculate the drag force F D of the longest setae. The advantage of this, is that it is then comparable appendage to appendage, and barnacle to barnacle, which varies dramatically. Leakiness is defined as the ratio of the mass flow rate of fluid that flows through the inner setae to the mass flow rate of fluid at the inlet of the box, as depicted in Fig. 2.
4. Results Computational fluid dynamic simulations revealed details of the flow that are difficult to observe on animals experimentally. Due to computational limitations and to the modular nature of barnacle appendages, only a small three-dimensional portion of the barnacle feeding appendages were modeled (the box in Fig. 1B is the central portion of cirri VI, which was used for CFD modeing). Observations on whole animals using the laser-illumination flow visualization technique provided a larger flow field, though with less resolution. In situations where results from CFD and experiments could be compared, there were several congruent findings.
This congruence provides confidence that our CFD simulations were accurate. 4.1. Flow dynamics Figs. 3 and 4 show the velocity contour in the XZ plane and around the cirri at three velocities (U ¼ 0:04; 0:21, and 0.50 m/s), for sheltered and exposed barnacles, respectively. Flow direction is from right to left. Figures on the top illustrate a twodimensional plane (XZ plane) through the longest (top) setae of a rami segment. Figures on the bottom illustrate single rami segments in 3D. For comparison, photographs of the analogous aerial view obtained experimentally, at lower Re, are shown in Fig. 5. At the low velocity, the boundary layers do not differ substantially between the sheltered (Fig. 3A) and exposed (Fig. 4A) models. There is about an equal amount of blue,1 which refers to low flow velocity. At high velocity, the amount of blue is greatest around the exposed model (Fig. 3C and Fig. 4C), demonstrating that the boundary layers are thicker. As Re increases, the boundary layer thickness decreases, which allowed flow to pass between the filtering structure faster. This would increase particle encounter with cirri and setae (Larsson and Jonsson, 2006). The fine details of the boundary layer were difficult to measure from the videos taken during the laser-illumination experiments, although a decrease in velocity directly adjacent to both barnacle surfaces was documented (SI Videos 1 and 2). Wake characteristics 1 For interpretation of color in Figs. 3 and 4, the reader is referred to the web version of this article.
232
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
and vorticity intensity strongly depend on the Reynolds number, which is determined by the phenotype, as deduced from laserillumination observations. When Re < 1, the fluid goes around the barnacle rami, and the boundary layers do not separate from the surface at any point. The resulting flow field, illustrated in Fig. 5A and D, is steady and also known as creeping flow. As Re increases, the flow begins to separate at the rear stagnation point and the flow pattern changes to the steady twin vortex regime where two steady symmetric vortices are formed behind the cirri (flip-flopping). At higher velocities, both the exposed and sheltered phenotypes, resulted in periodic vortex shedding. This was most prominent with the exposed phenotype, where vorticies deflected rapidly from side to side. 4.2. Drag force F D Drag force represents the hydrodynamic force that exerts a body due to fluid flow passing around it (Koehl, 1996), and is represented as:
FD ¼
1 qU 2 AC D 2
ð2Þ
with q the fluid density (kg=m3 ), U the free stream velocity (m=s), A the area at which the flow is exposed to (m2 ) and C D the drag coefficient. C D is a function of Re and highly depends on the size and shape of the object, which cannot easily be deduced from experiments, but was accurately obtained from CFD. C D was obtained for the longest setae; therefore A in Eq. (2) is the area of the longest setae (b s). This provided new insight into the functional morphology of the two barnacle phenotype models. Fig. 6 shows that the drag force for both exposed and sheltered models are the same for the Re studied, even though C D was quite different. Drag force is directly proportional to the exposed surface area of the animal. Although exposed barnacles have thicker structures, the shorter length (see Table 2) minimizes the surface exposed to oncoming flow, which reduces the drag force with respect to sheltered barnacles. On the other hand, exposed barnacles have a smaller spacing ratio (i=b) resulting in lower drag force
Fig. 6. Drag force F D versus Reynolds number Re for inner and outer setae.
because of the high interference of flow from neighboring setae due to a broad viscous region (larger boundary layer thickness). The small surface area and small spacing ratio of exposed barnacles trade off of each other, resulting in similar drag force as sheltered barnacles. Fig. 6A and B show that for both exposed and sheltered models, the drag force exerted to the inner setae is slightly higher than the outer setae at the same Re. This is due to the higher viscous layer around inner setae due to the interference of the neighboring setae from all directions. 4.3. Leakiness Leakiness is defined as the ratio of the mass flow rate of fluid that flows through the gap between the inner setae to the mass flow rate of fluid at the inlet of the box, indicated with green arrows in Fig. 2. Fig. 7 shows values of leakiness for a wide range of Re from our CFD results. At around Re 1, the sheltered barnacle model demonstrated nearly three times higher leakiness through the enclosed region compared to the exposed barnacle model. The flow around exposed model at the same Reynolds, showed the characteristics of a creeping flow, indicating little to no leakiness, like a paddle (Fig. 4A). The higher velocity in between gap spaces of filters observed in CFD velocity profiles may be interpreted as fluid passing between the segments and setae, which corresponds to a leaky state (Figs. 3C and 4C). As Re increased, the leakiness of filters increases for both models, with sheltered filters showing a higher percentage passing through overall. Due to experimental restriction, we could not quantify leakiness from the laser-illumination analysis. As a best proxy, leakiness was assumed once particles were observed to pass directly through the center of a filter, as depicted with arrows in Fig. 8. At the lowest velocity tested (i.e. 0:03 m=s; Re ¼ 0:43 for sheltered and Re ¼ 0:52 for exposed), both barnacle filters were non-leaky paddles. As velocity approached 0.062 m/s (Re ¼ 0:89), comparable to our lowest CFD velocity tested, sheltered barnacles revealed slight leakiness. This velocity is higher than that measured from the sheltered collection site (Table 1). Leakiness increased as the flow velocity was increased up to 0.16 m/s (Re ¼ 2:24), after which complete leakiness was observed (Fig. 8B and Video 1). Exposed barnacles revealed onset of leakiness at 0.14 m/s (Re ¼ 2:63) with full leakiness at 0.18 m/s (Re ¼ 3:50) (Fig. 8D and Video 2). In conclusion, there is a critical range of Reynolds numbers, beyond which full leakiness is observed (sieve-like behavior) and below which no leakiness is observed (paddle-like behavior). We refer to this critical range as the leaky-paddle transition range. The leaky-paddle range for sheltered and exposed barnacles are depicted in Fig. 7 as shaded areas and lie within the results
Fig. 7. Percent of mass flow rate (leakiness) passing through inner setae at increasing Re from CFD results. The shaded areas are critical values obtained from laser illumination experiments. For Re values smaller than the critical range, leakiness was not observed (paddle-like behaviour), and for Re values larger than the critical range leakiness was observed.
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
233
Fig. 8. Laser-illumination experiments illustrating sieve-like behavior in sheltered and exposed barnacle filters with flow passing from right to left. A–C show sieve-like behavior for sheltered barnacles at U ¼ 0:16 m=s (Re ¼ 2:24). D–F show sieve-like behavior for exposed barnacles at U ¼ 0:18 m=s (Re ¼ 3:50). White arrows denote movement of a particle passing directly through the center of filter.
obtained by CFD. Our CFD and experimental results showed that leakiness for the sheltered phenotype initiates at lower Re. The transition from paddle to leaky occurs over a narrower Re range for the exposed phenotype (Fig. 7).
5. Discussion Balanus glandula has remained at the centre of marine biology investigations since Darwin (1854). The extreme modification of ramus and seta length in response to water velocity is one of the best documented ecophenotypes (Marchinko and Palmer, 2003). The majority of cirrus variation observed in nature arises from environmentally induced phenotypic plasticity (Arsenault et al., 2001; Marchinko and Palmer, 2003). The development of rami form is clearly correlated with flow velocity. Here we quantify leakiness as a function of Reynolds number (see Eq. (1)). We show that shorter, stouter feeding rami (exposed models) have the same drag force to longer and slimmer feeding rami (sheltered models) because of their smaller spacing ratio, but the range of flow velocity over which feeding appendages transition from paddle to sieve is differs between models. This comparative approach revealed details of flow in otherwise hard to see regions of B. glandula feeding rami, and revealed flow details at high flow velocities that have proven impossible to observe. This comparative approach could equally be used to compare different periods of an individuals growth, or to quantify interspecific differences at one developmental period. We chose to quantify an intraspecific phenotypic difference, because like interspecific comparisons, it is an example of developmental repatterning, but it is unique in that it sheds light on the critical marine ecological parameter, flow velocity. Another variable that affects the Reynolds number is temperature. Warmer water is less viscous. Balanus glandula experiences seasonal and latitudinal ranges in temperature. It geographic range is from Alaska to the Baja Peninsula. Temperature-induced viscosity change likely impact the feeding biology including food detection, capture, and ingestion rates (Podolsky, 1994), and it has a dramatic effects on the physiology of B. glandula (Nishizaki and Carrington, 2015).
At low Re the velocity gradient around feeding appendages produces a large viscous displacement effect, which diverts the flow away from surfaces due to the broad viscous region, consequently pushing particles away and making the filter less effective as a sieve to strain particles (Koehl and Strickier, 1981; Cheer and Koehl, 1987a). The critical range for transitioning from a paddle to a sieve for exposed Balanus glandula filters occurs between 2:63 < Re < 3:5 (0:14 < Uðm=sÞ < 0:18), while sheltered barnacles have a lower threshold of 0:89 < Re < 2:24 (0:062 < Uðm=sÞ < 0:16) for sieving. Within the critical transitioning range, the two phenotypes function very differently. The longer, slimmer geometry of sheltered barnacles provides more space for the flow to pass. Interestingly, we found that the paddle to sieve transition occurred at a higher velocity (0:062 < Uðm=sÞ < 0:16) than the maximum velocity reported from the sheltered, Grappler Narrows, collection site (0.0052 m/s, Table 1) (Marchinko and Palmer, 2003). Theoretically, the leakiness will increase feeding efficiency; but in reality, barnacles from very sheltered habitats may not experience leakiness, or must actively feed to achieve leakiness. Reasons why sheltered B. glandula do not sieve, may be ecological - a high food density in Grappler Narrows may negate the need for a sieve - or mechanical - longer legs may be more prone to breakage at the high velocities. The shorter, stout, geometry of exposed barnacles reduce drag forces, and transition to a sieve at high flow velocities, experienced in the field. The average maximum velocity in Seppings Island site is 4.41 m/s (Marchinko and Palmer, 2003), which is higher than the leaky-paddle transition for exposed barnacles obtained from our CFD results, meaning that exposed barnacles in the Seppings Island site function as sieves. The high velocity and continuous influx of particle-filled water passing through exposed filters provide a larger volume of flow per unit surface. This allows them to present less surface to the flow while maintaining the capacity to efficiently filter particles. The wake that develops downstream of barnacle feeding appendages gives information about the leakiness. This wake is controlled by Reynolds number and gap spacing (Kang, 2003). A large, slow-moving wake with counter-rotating eddies was observed from laser-illumination experiments and also in CFD simulations for both the exposed and sheltered barnacles. These wakes
234
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
are comparable to wakes formed by flow over solid bluff objects (Kang, 2003; Meneghini et al., 2001; Zhang and Zhou, 2001). At low Re, little flow passes between hairs resulting in a paddle-like function. To validate this result, a new set of simulations were performed (Fig. 9). A two-dimensional CFD simulation comparing flow through 6 uniform diameter cylinders arranged in a side-by-side arrangement normal to flow versus an array of 6 cylinders of decreasing diameters at Re ¼ 0:5 revealed similar large recirculating vortices behind segments for the uniform cylinder arrangement (Fig. 9). This indicates that uniform cylinders row is acting as a bluff object at this low Re. However, increasing the relative center to center distance to diameter ratio, allows the flow to pass through at the same Reynolds number value. Increasing the gap to diameter ratio is analogous to switching from an exposed to a sheltered barnacle. This effect evidences the ability for barnacles to adapt biological filters to the flow environment. At low Re, when cylinders are very packed, the boundary layers interfere, and acts as a wall that blocks flow between them (Kang, 2003; Meneghini et al., 2001; Zhang and Zhou, 2001). At the lower range of tested velocities, for Re lower than 1, the difference in setae spacing ratio (i=b) has little effect on the flow rate through the appendage, both behave as paddles. This is similar to what is observed in cylinder interference observations. When i=b is lower than 3, both for the exposed and sheltered barnacles (Table 2), at these low Reynolds numbers values (i.e. Re < 1), filters act as a
single bluff body keeping gap flow minimal and deflecting the flow to the outside of the appendages. As flow velocity increases, twin recirculating vortices observed with the laser-illumination experiments, for both phenotypes, indicated more gap flow rate (i.e. flow passing through the appendages) which in turn influenced wake patterns (Videos 1 and 2). The deflection of the gap flow was irregular and unsteady, resulting in a narrow and a wide vortex street. As flow separation and flow between cirri was not yet evident, gap flows were assumed to be weak, therefore the filters behave as slightly leaky paddles. As flow velocity increased further, for Re ¼ 2:24 (sheltered) and 3.5 (exposed), wakes with flow separation points located further rearward from cirri indicated that a steady flow passed through gap spaces, indicating full leakiness of the filters. Our CFD results and experimental observations are congruent with respect to general flow and wake patterns. However, limitations with the numerical models have to be stressed. All simulations were performed on passive barnacle filters, and active feeding behavior was not considered. This is a limitation because the movement of an appendage relative to fluid and towards a wall (i.e. substrate, body surface, or neighbor) can change the leakiness at low Re (Cheer and Koehl, 1987b; Koehl, 1996; Loudon et al., 1994). As barnacles are facultative suspension feeders, actively sweeping their filters in slow flow and passively extending filters held rigidly opposed to the direction of flow in faster moving water, sheltered barnacles would normally actively feed at the lower end of our velocities tested. Similarly, changes in setae spacing ratio (i=b) may affect the leakiness of a filter moving relative to a wall (Loudon et al., 1994). On the other extreme, computational simulations may overestimate drag coefficients because models are fully rigid. They do not bend and thus present the same surface to the flow for all velocities. Our laser-illumination observations, and Marchinko and Palmer (2003), Ferner and Gaylord (2008), show that sheltered barnacles undergo major cirral deformation for flow velocities above 0.21 m/s, while exposed cirral fans undergo minor cirral deformation at 0.49 m/s. The two simulation models represent only 6 medially located rami segments (Fig. 9), to keep computational time and calculations reasonable. This enabled us to analyze both inner, interdigitated setae, and outer setae that are open to flow. A complete cirral filter under a range of Re may be expected to provide more realistic estimates on the transition from paddle to sieve. Another limitation with the numerical models was that the feeding appendages in the model were assumed smooth elliptical cylinders, while, arthropod exoskeletons are composite materials that have complex structural reactions to flow with surfaces that are typically not smooth. Despite these limitations, these CFD simulations, validated against direct observations, show that Balanus glandula from exposed sites are remarkably adaptable to a wide range of flow velocities, whereas those from sheltered sites may not be mechanically optimal. Their abundance in sheltered bays however, suggest there whatever the reason, ecological, developmental, or mechanical, the trade-off is a good one. The beauty of biological fluid mechanics, is that the physical selection pressure can be precisely quantified, and the fluid forces have not changed throughout the origin and evolution of life. Conflict of interest statement The authors have nothing to disclose. Acknowledgments
Fig. 9. Velocity contours through (A) uniform cylinders and (B) cylinders of decreasing size.
Thank you to E. Tixier, R. Yu, and S. Corbeil-Letourneau at Ecole Polytechnique for discussions and assistance with finite volume
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
models, and CFD, and to the director and staff of the Bamfield Marine Sciences Centre for logistic support. Conversations with A.R. Palmer (University of Alberta) were central to our decision to embark on this project. We are grateful to Jason Hodin, Mike Nishizaki and one anonymous reviewer for improvements to this publication. This project was funded by an NSERC Discovery grant to C.B. C. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.jbiomech.2018.02. 011. References Arsenault, D.J., Marchinko, K.B., Palmer, A.R., 2001. Precise tuning of barnacle leg length to coastal wave action. Proc. R. Soc. London, Ser. B 268 (1481), 2149– 2154. Cheer, A.Y.L., Koehl, M.A.R., 1987a. Fluid flow through filtering appendages of insects. Math. Med. Biol. 4 (3), 185–199. Cheer, A.Y.L., Koehl, M.A.R., 1987b. Paddles and rakes: fluid flow through bristled appendages of small organisms. J. Theor. Biol. 129 (1), 17–39. Darwin, C., 1854. A Monograph on the Sub-Class Cirripedia, With Figures of all the Species, vol. 1. Ray society, London. URL. Ferner, M.C., Gaylord, B., 2008. Flexibility foils filter function: structural limitations on suspension feeding. J. Exp. Biol. 211 (22), 3563–3572. Geierman, C., Emlet, R., 2009. Feeding behavior, cirral fan anatomy, reynolds numbers, and leakiness of Balanus glandula, from post-metamorphic juvenile to the adult. J. Exp. Mar. Biol. Ecol. 379 (1), 68–76. Hansen, B., Tiselius, P., 1992. Flow through the feeding structures of suspension feeding zooplankton: a physical model approach. J. Plank. Res. 14 (6), 821–834. Happel, J., Brenner, H., 2012. Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, vol. 1. Springer Science & Business Media. Jorgensen, C., 1983. Fluid mechanical aspects of suspension feeding. Mar. Ecol. Prog. Ser. 11, 89–103. Kang, S., 2003. Characteristics of flow over two circular cylinders in a side-by-side arrangement at low reynolds numbers. Phys. Fluids 15 (9), 2486–2498. Koehl, M.A.R., 1993. Hairy little legs: feeding, smelling, and swimming at low Reynolds numbers. Contemp. Math 141, 33–64. Koehl, M.A.R., 1996. When does morphology matter? Annu. Rev. Ecol. Evol. Syst. 27 (1), 501–542. Koehl, M.A.R., 2001. Transitions in function at low Reynolds number: hair-bearing animal appendages. Math. Methods. Appl. Sci. 24 (17–18), 1523–1532.
235
Koehl, M.A.R., Strickier, J.R., 1981. Copepod feeding currents: food capture at low Reynolds number. Limnol. Oceanogr. 26 (6), 1062–1073. LaBarbera, M., 1984. Feeding currents and particle capture mechanisms in suspension feeding animals. Am. Zool. 24 (1), 71–84. Larsson, A.I., Jonsson, P.R., 2006. Barnacle larvae actively select flow environments supporting post-settlement growth and survival. Ecology 87 (8), 1960–1966. Loudon, C., Best, B., Koehl, M.A.R., 1994. When does motion relative to neighboring surfaces alter the flow through arrays of hairs? J. Exp. Biol. 193 (1), 233–254. Marchinko, K.B., 2007. Feeding behavior reveals the adaptive nature of plasticity in barnacle feeding limbs. Biol. Bull. 213 (1), 12–15. Marchinko, K.B., Geary, D., 2003. Dramatic phenotypic plasticity in barnacle legs (Balanus glandula Darwin): magnitude, age dependence, and speed of response. Evolution 57 (6), 1281–1290. Marchinko, K.B., Palmer, A.R., 2003. Feeding in flow extremes: dependence of cirrus form on wave-exposure in four barnacle species. Zoology 106 (2), 127–141. Mehrabian, S., Letendre, F., Cameron, C.B., 2018. The mechanisms of filter feeding on oil droplets: theoretical considerations. Mar. Environ. Res., doi:https://doi.org/ 10.1016/j.marenvres.2018.01.006 (in press). Meneghini, J., Saltara, F., Siqueira, C., Ferrari, J., 2001. Numerical simulation of flow interference between two circular cylinders in tandem and side-by-side arrangements. J. Fluids Struct. 15 (2), 327–350. Miller, L.P., 2007. Feeding in extreme flows: behavior compensates for mechanical constraints in barnacle cirri. Mar. Ecol. Prog. Ser. 227. Nishizaki, M.T., Carrington, E., 2015. The effect of water temperature and velocity on barnacle growth: quantifying the impact of multiple environmental stressors. J. Therm. Biol. 54, 37–46. Podolsky, R.D. et al., 1994. Temperature and water viscosity: physiological versus mechanical effects on suspension feeding. Science 265, 100–103. Riisgård, H.U., Larsen, P.S., 2010. Particle capture mechanisms in suspensionfeeding invertebrates. Mar. Ecol. Prog. Ser. 418, 255–293. Rubenstein, D.I., Koehl, M.A.R., 1977. The mechanisms of filter feeding: some theoretical considerations. Am. Nat. 111 (981), 981–994. Schneider, C.A., Rasband, W.S., Eliceiri, K.W., 2012. NIH image to ImageJ: 25 years of image analysis. Nat. Meth. 9 (7), 671. Shimeta, J., Jumars, P.A., 1991. Physical mechanisms and rates of particle capture by suspension-feeders. Oceanogr. Mar. Biol. Annu. Rev 29 (19), l-257. Southward, A., 1955. On the behaviour of barnacles ii. the influence of habitat and tide-level on cirral activity. J. Mar. Biol. Assoc. U.K. 34 (03), 423–433. Tamada, K., Fujikawa, H., 1957. The steady two-dimensional flow of viscous fluid at low Reynolds numbers passing through an infinite row of equal parallel circular cylinders. Trager, G.C., Hwang, J.S., Strickler, J.R., 1990. Barnacle suspension-feeding in variable flow. Mar. Biol. 105 (1), 117–127. Vogel, S., 2003. Comparative Biomechanics: Life’s Physical World. Princeton, New Jersey, USA. Vogel, S., Savage, A.A., 1996. Life in moving fluids. The physical biology of flow. Biol. J. Linn. Soc. 57 (3), 291. Zhang, H.J., Zhou, Y., 2001. Effect of unequal cylinder spacing on vortex streets behind three side-by-side cylinders. Phys. Fluids 13 (12), 3675–3686.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
The fluid dynamics of Balanus glandula barnacles: Adaptations to sheltered and exposed habitats Maureen Vo a, Sasan Mehrabian a, Fernando Villalpando b, Stephane Etienne b, Dominique Pelletier b, Christopher B. Cameron a,⇑ a b
Departement de sciences biologiques, Universite de Montreal, Montreal, Quebec H3C 3J7, Canada Ecole Polytechnique de Montreal, C.P. 6079, succ. Centre-ville, Montreal, Quebec H3C 3A7, Canada
a r t i c l e
i n f o
Article history: Accepted 6 February 2018
Keywords: Balanus glandula Barnacles Leakiness Feeding Computational fluid dynamics
a b s t r a c t Suspension feeders use a wide range of appendages to capture particles from the surrounding fluid. Their functioning, either as a paddle or a sieve, depends on the leakiness, or amount of fluid that passes through the gaps between the appendages. Balanus glandula is the most common species of barnacle distributed along the Pacific coast of North America. It shows a strong phenotypic response to water flow velocity. Individuals from exposed, high flow sites have short and robust cirral filters, whereas those from sheltered, low velocity sites have long, spindly appendages. Computational fluid dynamics (CFD) simulations of these two ecophenotypes were done using a finite volume method. Leakiness was determined by simulating flow velocity fields at increasing Reynolds numbers, results that have been unattainable at higher velocities by observation. CFD also allowed us to characterize flow in hard to see regions of the feeding legs (rami). Laser-illumination experiments were performed at low to medium flow velocities in a flume tank and corroborated results from CFD. Barnacle filters from a sheltered site become completely leaky at Re ¼ 2:24 ð0:16 m=sÞ, well above the maximum habitat velocity, suggesting that this ecophenotype is not mechanically optimized for feeding. Barnacles from exposed environments become fully leaky within the range of habitat velocities Re ¼ 3:50 ð0:18 m=sÞ. Our CFD results revealed that the drag force on exposed barnacles feeding appendages are the same as the sheltered barnacles feeding appendages despite their shape difference and spacing ratio. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Suspension feeders use a diverse range of filamentous appendages to capture particles from the surrounding water (Mehrabian et al., in press; Riisgård and Larsen, 2010). Most appendages essentially consist of arrays of bristles or cylinders, and some behave as sieves to intercept particles from suspension passing through (e.g. bands of cilia, arthropod setae, and mucous nets), while others create feeding currents or act as paddles to direct pockets of fluid for further processing (e.g. flagella, cirri, or tentacles) (Riisgård and Larsen, 2010; Vogel, 2003). The performance of these appendages (i.e. paddle vs. sieve) depends on their interactions with the surrounding medium. Fluid can either pass readily through the gaps between filaments, creating a ‘leaky’ appendage that functions as a sieve, or it may flow around the perimeter of the appendage that functions as a paddle.
⇑ Corresponding author. E-mail address: [email protected] (C.B. Cameron). https://doi.org/10.1016/j.jbiomech.2018.02.011 0021-9290/Ó 2018 Elsevier Ltd. All rights reserved.
The feeding behaviour and flow about several species of barnacles have been described, especially in Balanus glandula (Geierman and Emlet, 2009; Marchinko, 2007; Marchinko and Palmer, 2003; Miller, 2007; Southward, 1955; Trager et al., 1990). B. glandula (Fig. 1) is a common Northeast Pacific species of acorn barnacle found in a wide range of wave-exposed habitats. It has been subject to questions about how morphology and fluid dynamics affect appendage functioning. Its feeding legs demonstrate a great degree of phenotypic plasticity in response to water velocity (Marchinko, 2007; Marchinko and Palmer, 2003; Southward, 1955). Its sessile nature facilitates easy observations and manipulation in the field or lab (Arsenault et al., 2001). The parameters of the feeding structures, including rami length and width, and seta length and spacing, of individuals found at low and high flow velocities is well characterized (Arsenault et al., 2001; Marchinko and Palmer, 2003). Less well known are how the water and appendage interact in confined, small, and hard to see regions close to the appendages. Also, observations at realistic, high flow velocities have not been
226
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
Fig. 1. Microscopic image of the entire feeding appendage of A: Exposed and B: Sheltered Balanus glandula. I, II, III, IV, V, and VI refer to the cirrus numbers. Each cirrus consists of 2 rami. The box shows the portion of cirri VI used for CFD modeling. C: Microscopic image of a ramus and segments of a feeding appendage. D: Frontal view of the two rami of cirrus VI. E: Schematic front view of a ramus segment. h is the length of the segment, w is segment width, b is the setae diameter, i is the centre-to-centre setae spacing, s is the setae length, and l is segment length, which is in the flow direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
reported in the literature. Several mathematical and physical models of fluid velocity profiles around rows of cylinders of finite width have demonstrated that the transition to leakiness depends largely on the Reynolds number and setae spacing ratio (Cheer and Koehl, 1987a,b; Hansen and Tiselius, 1992; Koehl, 1993; Loudon et al., 1994; Mehrabian et al., in press; Tamada and Fujikawa, 1957). The Reynolds number represents the ratio of inertial to viscous forces:
Re ¼ Ub=m
ð1Þ
where U is the free flow velocity, b the characteristic length, in our case the base diameter of the longest setae (see Fig. 1E) and m the kinematic viscosity of the surrounding fluid. For an incompressible isothermal flow, this number fully characterizes the fluid flow around a given geometry pattern, like a filament (Vogel, 2003). For most biological filaments, the Reynolds number ranges from
105 to 10 (Cheer and Koehl, 1987b; Jorgensen, 1983; LaBarbera, 1984; Rubenstein and Koehl, 1977; Shimeta and Jumars, 1991). At low values, viscous forces dominate, velocity boundary layers are thick relative to the dimensions of the body, and the flow about the appendage is laminar (Happel and Brenner, 2012; Vogel and Savage, 1996). But, from dimensional analysis, when geometry varies, geometrical ratios have to be considered. For example, the setae spacing ratio, i=b (the ratio of setae center-to-center spacing, i, to setae diameter b, see Fig. 1E) increases from sheltered to exposed barnacles and is known to have an important effect on leakiness (Marchinko and Palmer, 2003). In general, for Re ¼ 0:5, high leakiness is observed when i=b becomes larger than 5 (Cheer and Koehl, 1987b; Koehl, 1996, 2001). However, for Re of the order of 103 , changes to this ratio has little influence on leakiness (i.e. morphological and behavioural diversity without performance consequences) (Koehl, 1996).
227
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
The mechanisms and rates of particle capture by filter feeding appendages depend on the flow field around the individual filaments of a filter (Rubenstein and Koehl, 1977; Shimeta and Jumars, 1991). Specific morphological changes and novel functions can result simply from changes in size, shape, stiffness, or habitat of an organism. To understand the relationships between morphology and functioning and the basic physical rules governing how a biological structure operates, we embarked on a mechanical study. The objectives of this study are to compare the flow dynamics, forces (drag) on the barnacle cirri, and the leakiness, between passive feeding barnacles from exposed (i.e. robust phenotype) and sheltered (i.e. spindly phenotype) shores across a range of realistic flow velocities. In this paper, we study the appendage-water interaction for the robust, and spindly barnacle cirri phenotypes. We visualize these poorly known interactions, and characterize them using dimensionless numbers, in low-velocity, medium velocity, and high velocity scenarios for the spindly phenotype, and the robust phenotype. We accomplish these goals using a computational fluid dynamics (CFD) tool, and validate our results by comparing our findings to observations of B. glandula exposed to flow at different velocities experimentally in a flume tank. The information gained from this comparative approaches (CFD model, and observations) is significant because it precisely quantifies the amount of force applied to the appendages and the amount of water that flows between the appendages (leakiness) at different flow velocities. Also, we are able to visualize the flow behavior around B. glandula rami across a range of realistic velocities and determine the velocities at which each phenotype transitions from paddle to ‘leaky’ sieve. 2. Experimental materials and methods 2.1. Collection sites Balanus glandula specimens were collected from two sites in Barkley Sound, Vancouver Island, British Columbia, Canada
between May and July, 2013 (Table 1). Rocks and mussels (Mytilus californianus) with attached barnacles were collected from the sheltered and low flow shoreline of Grappler Inlet, and from the wave exposed and high flow shoreline of Seppings Island (Table 1). Careful attention was paid to choose barnacles from microniches that were not obviously obstructed by adjacent obstacles. These collection locations are the same as those used by Marchinko and Palmer (2003). Table 1 reflects average maximum velocity rates recorded by Marchinko and Palmer (2003). Barnacles were transported to the Bamfield Marine Sciences Centre and maintained in sea water Table (181 cm long 84 cm wide 8 cm deep). Barnacle specimens used for fixing, measuring and testing were used immediately upon collection or within the first two days. 2.2. Fixing of barnacles Barnacle specimens were removed from their shells and placed in a 7% solution of MgCl2 to relax the feeding legs (rami). Specimens were then transferred to Bouin’s fixative for 24 h to fix legs in the open, passive feeding position and stored in 70% ethanol. 2.3. Cirral anatomy measurements As phenotype varies widely in B. glandula, the goal was to characterize the two drastically different shapes for flow observations and CFD analysis that represent sheltered and exposed barnacles. Six barnacle samples from exposed environments, and six barnacles from sheltered environments, were used to obtain averages for the parameters used to construct the CFD model (see Table 2). The posterior three cirri (IV, V, VI) were selected since they are the longest and primary food capturing appendages (Marchinko and Palmer, 2003). Individual barnacles were observed with an Olympus Q Color 5 camera mounted on an Olympus SZX16 dissecting microscope. Images, similar to those shown in Fig. 1, were captured using Q capture software (Surrey, BC, Canada). The cirrus with the longest rami, cirrus VI, was removed, straightened into the feeding position, and the ventral surfaces and side profiles were
Table 1 Collection sites and descriptions. Wave exposure
Site
Sheltered Exposed ⁄ ⁄⁄
Grappler narrows Seppings Island
Location description
North latitude⁄⁄ 0
West-facing shores Western-most rocks on southwest facing
00
48°49 91 48°500 5000
West latitude⁄⁄ 0
Water velocity⁄ (m/s)
00
125°07 64 120°120 5000
0.0052 4.41
Adapted from Marchinko and Palmer (2003). Barnacles collected from Barkley Sound, Vancouver Island, British Columbia, Canada.
Table 2 Summary of sheltered and exposed barnacle dimensions. Each data point is an average of n ¼ 6 barnacle samples. Barnacle type
Setae
Segment
Setae number⁄
Length s (lm)
Base diameter b (lm)
Tip diameter t (lm)
Spacing ratio i=b
Angle h (deg)
Length l (lm)
Height h (lm)
Width w (lm)
Sheltered (n = 6)
1 2 3 4 5 6
515 58 363 39 224 21 147 17 90 11 62 12
15 4 16 3 13 2 11 2 81 51
4 0:78 4 0:78 2 0:50 2 0:50 2 0:50 0:5 0:20
1.50 1.69 1.71 1.82 2.29 3.00
47 2
186 13
216 19:1
93 19
Exposed (n = 6)
1 2 3 4 5 6
445 14 314 87 275 68 161 64 106 28 65 20
20 2 20 2 18 1 15 2 10 2 6 1:5
5 1:30 5 0:82 4 0:78 3 1:30 3 0:79 2 0:50
1.20 1.08 1.09 1.25 1.45 2.50
51 3
280 44
164 24
98 8
is the standard deviation of the averaged data points. ⁄ Each setae number refers to an individual setae as illustrated in Fig. 1E.
228
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
photographed (Fig. 1A–D). Morphometric analysis using ImageJ (developed by the National Institutes of Health, Bethesda, MD, USA) of enhanced micrographs and scanning electron microscope images of central segments of cirri VI produced values for segment length, width, and height, setae lengths and widths (at the proximal and distal ends), angle of setae with respect to the cirri segments, intersetal spaces, and distance between cirri (Fig. 1A–C, Table 2).
2.4. Laser-illumination flow visualization Flow visualization was performed in a laboratory flume having 107 cm length, 18 cm width and 19 cm depth (see Supporting Fig. 1), at a water temperature of 12 °C. Extracted barnacle bodies were held 70 cm from the start of the chamber with tweezers embedded in modelling clay, such that the feeding legs were free to open in the passive feeding position, in a mid-water position
Fig. 2. GAMBIT models illustrate front and top view of sheltered and exposed barnacles. Computational model of six cirral segment and setae of B. glandula with boxed domain boundary, showing the direction of flow (green arrows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
of the flume. A class IIIb, wavelength 532 nm laser (Laserglow technologies) was passed through a concave crystal to create a two dimensional horizontal sheet across the mid-section of the barnacles cirri for flow field observations. The flume water was seeded with silver-coated neutrally buoyant hollow glass particles of 11 lm in diameter (110P8, LaVision GmbH) at a concentration of 10:50 g=ml. The particles were neutrally buoyant, and therefore motion was only horizontal, in the direction of flow. Fully developed flow fields around filters were recorded using a Sony HDV (1080i HVR-Z1U/Z1N, 60i) video recorder at a rate of 30 frames per second. Water velocity was determined by frame-byframe video analysis of suspended particles across a distance of 20 cm, taken 10 cm before the passively positioned barnacles. Only a few particles, that could be reliably tracked, were used to establish flow speeds, which were constant. At times, particle visualization was improved by injecting a high density of particles directly upstream of the feeding legs using a pipette. A variable speed motor (90 volt, 7.6 amp) and propeller was used to control flow velocity that ranged from 0.03 m/s to 0.19 m/s. Leakiness was determined once particles were observed to pass directly through the barnacle appendages. Like those who studied barnacle appendage-water interaction systems before us, we were unable to see flow in confined spaces, nor at high velocities. Also at Reynolds numbers higher than 6, particles motions were too fast to evaluate velocity vectors accurately. iMovie ’11 (version 9.0.4) and ImageJ (Schneider et al., 2012) were used to analyze the video footage. 3. CFD model and fluid flow simulations In this section, we carried out numerical simulations of the flow around outer rami, which are exposed to free flow, and between opposing rami, where setae interdigitate, for the two phenotypes (sheltered and exposed). Our objectives were to (1) visualize flow pathlines, and velocity vectors/contours in confined spaces of the barnacle appendages, (2) to obtain the drag force applied to the appendages at various Re, and (3) to simulate realistic flow conditions to determine the Re at which the appendages transition from paddle to sieve. The accuracy of our simulations were verified against our laser-illumination visualizations. 3.1. Three-dimensional models Each cirrus has two rami (legs). Six segments of one and a half rami were modeled as elliptical cylinders (Fig. 2). The segments vary in height, width, and length characteristic of each barnacle phenotype (Table 2). All geometries were converted to dimensionless form, to facilitate scale-up of obtained results to real conditions and to simplify and generalize the numerical calculations. The base diameter of the longest setae from the sheltered barnacle was used as the reference number to divide all parameters of both models. This converted all our models from dimensional to dimensionless scale for simulations. Projecting anterior-laterally from each segment were six pairs of conical-shaped setae, which protruded from the segments at an angle of 47 for the sheltered model, and 51 for the exposed one, with respect to the corresponding pair (Fig. 1D). Distances of 500 lm between sheltered rami, and 400 lm between exposed rami were chosen for the CFD model. These distances are realistic, and represent the smaller spacing of the exposed phenotype. Specific segment parameters are summarized in Table 2. As illustrated in Fig. 1, the distal two setae interdigitate with those of the adjacent rami, whereas a gap was modeled between the shorter, more basal, setae. In order to minimize memory requirements and computational time, six segments are modeled, and one of the rami (elliptical cylinders)
229
was divided in half and reflected as a mirror image using the ’symmetrical’ function when defining boundary conditions. The other segment was left whole to examine flow behavior between the interdigitating setae, and around the outer setae that was exposed to a free flow stream (Fig. 2). 3.2. Meshing The 3D spatial discretization of the two barnacle filter models were generated using GAMBIT (Fluent 13, ANSYS, 2012, Cecil Township, Pennsylvania, USA) a Fluent CFD pre-processing program. A variety of verification tests were performed to assess the appropriate number of cells and the fluid domain size to use in the CFD simulations. A width of 80 diameters of the reference model, which is the base diameter of the longest setae from the sheltered barnacle, was found to have negligible blockage effect. Three mesh sizes of approximately 3, 7 and 9 million cells, standing for coarse, medium, and fine meshes respectively were tested on each of the barnacle models. The mass flow rate at the segment was monitored for each case and a final mesh size at which increasing the grid size showed no significant changes in mass flow rates was found to be 6,808,487 cells for the sheltered barnacle model and 7,439,983 cells for the exposed one. More specifically, each setae tip and base is defined with 20 nodes around the circumference and nodes along the length of setae ranging from 250 to 45 from longest to shortest setae, respectively. All setae and the region directly surrounding setae are areas of high interest to understand the feeding mechanics of B. glandula. Therefore, small cells were inserted in this region. Boxed areas were created around the filter structures to control the cell size in critical areas (Fig. 2). A size function using setae as the source of growth allowed cell size to increase farther away from the source at a growth rate of 1.05. 3.3. Boundary conditions The steady-state incompressible Navier-Stokes equations which represent conservation of mass and momentum, can be solved provided that boundary conditions are prescribed on all surfaces surrounding the fluid domain. The boundary conditions imposed in these simulation were as follows: Inlet conditions: constant horizontal speed (uniform velocity profile). Outlet conditions: pressure outlets. Left wall: symmetry conditions. Right wall: wall boundary condition (no-slip boundary condition). Barnacle/water interface: wall boundary condition (no-slip boundary condition). The fluid is seawater, and was considered as incompressible with a density of 1024 kg/m3 and a kinematic viscosity m of 1:047 106 m2 =s at 20 °C (Vogel and Savage, 1996). 3.4. CFD model The commercial CFD software package ANSYS FLUENT (version 13.0) based on a finite volume method was used to solve the threedimensional Navier-Stokes equations. In this method, the field of calculation is divided into a certain number of meshes, called control volumes. The system of equations governing the flow (continuity and momentum equations) is discretized at the level of these meshes, which leads to a system of non-linear algebraic equations solved in an iterative way.
230
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
Our simulations were carried out for three flow velocities (U ¼ 0:04; 0:21 and 0:50 m=s). These inlet velocities correspond to the flow velocity range used in previous B. glandula feeding studies (Marchinko, 2007; Marchinko and Geary, 2003), and measured in situ (Geierman and Emlet, 2009; Trager et al., 1990). Also, the velocities tested in the laser illumination experiments are within
the lower end of this range (i.e. U ¼ 0:03—0:19 m=s). Furthermore, given the small size of the barnacle filters (see Table 2) and the flow velocities considered, the Reynolds number defined in Eq. (1) is of the order of 101 to 100 . This implies that the flow around the barnacle filters is laminar. As a consequence, the SIMPLE algorithm, QUICK momentum scheme, ‘Green-Gauss cell
Fig. 3. Top view of velocity contours and magnified velocity vectors passing through intersecting setae with the direction of flow from right to left for sheltered barnacles. Green arrows indicate flow direction. Velocities in the scale bar are normalized with respect to the free stream velocity U. To calculate the real velocities multiply these numbers by the inlet velocity.
Fig. 4. Top view of velocity contours and magnified velocity vectors passing through intersecting setae with the direction of flow from right to left for exposed barnacles. Green arrows indicate flow direction. Velocities in the scale bar are normalized with respect to the free stream velocity U. To calculate the real velocities multiply these numbers by the inlet velocity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
231
Fig. 5. Laser-illumination of water flow and wake behavior viewed from above a passive B. glandula feeding appendage. The direction of flow is from right to left. A–C is a sheltered barnacle and D–F is exposed.
based’ gradient and pressure-velocity coupling method were used to solve the problem. The sum of normalized residuals of all variables were considered converged when lower than 104 . This occurred within 900–3000 iterations for all models tested. Our CFD results were quantified based on two parameters: drag force F D and leakiness. Drag force is the sum of the stresses and pressure over the surface area of the feeding appendage. In this study, the drag coefficient C D was obtained from FLUENT to calculate the drag force F D of the longest setae. The advantage of this, is that it is then comparable appendage to appendage, and barnacle to barnacle, which varies dramatically. Leakiness is defined as the ratio of the mass flow rate of fluid that flows through the inner setae to the mass flow rate of fluid at the inlet of the box, as depicted in Fig. 2.
4. Results Computational fluid dynamic simulations revealed details of the flow that are difficult to observe on animals experimentally. Due to computational limitations and to the modular nature of barnacle appendages, only a small three-dimensional portion of the barnacle feeding appendages were modeled (the box in Fig. 1B is the central portion of cirri VI, which was used for CFD modeing). Observations on whole animals using the laser-illumination flow visualization technique provided a larger flow field, though with less resolution. In situations where results from CFD and experiments could be compared, there were several congruent findings.
This congruence provides confidence that our CFD simulations were accurate. 4.1. Flow dynamics Figs. 3 and 4 show the velocity contour in the XZ plane and around the cirri at three velocities (U ¼ 0:04; 0:21, and 0.50 m/s), for sheltered and exposed barnacles, respectively. Flow direction is from right to left. Figures on the top illustrate a twodimensional plane (XZ plane) through the longest (top) setae of a rami segment. Figures on the bottom illustrate single rami segments in 3D. For comparison, photographs of the analogous aerial view obtained experimentally, at lower Re, are shown in Fig. 5. At the low velocity, the boundary layers do not differ substantially between the sheltered (Fig. 3A) and exposed (Fig. 4A) models. There is about an equal amount of blue,1 which refers to low flow velocity. At high velocity, the amount of blue is greatest around the exposed model (Fig. 3C and Fig. 4C), demonstrating that the boundary layers are thicker. As Re increases, the boundary layer thickness decreases, which allowed flow to pass between the filtering structure faster. This would increase particle encounter with cirri and setae (Larsson and Jonsson, 2006). The fine details of the boundary layer were difficult to measure from the videos taken during the laser-illumination experiments, although a decrease in velocity directly adjacent to both barnacle surfaces was documented (SI Videos 1 and 2). Wake characteristics 1 For interpretation of color in Figs. 3 and 4, the reader is referred to the web version of this article.
232
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
and vorticity intensity strongly depend on the Reynolds number, which is determined by the phenotype, as deduced from laserillumination observations. When Re < 1, the fluid goes around the barnacle rami, and the boundary layers do not separate from the surface at any point. The resulting flow field, illustrated in Fig. 5A and D, is steady and also known as creeping flow. As Re increases, the flow begins to separate at the rear stagnation point and the flow pattern changes to the steady twin vortex regime where two steady symmetric vortices are formed behind the cirri (flip-flopping). At higher velocities, both the exposed and sheltered phenotypes, resulted in periodic vortex shedding. This was most prominent with the exposed phenotype, where vorticies deflected rapidly from side to side. 4.2. Drag force F D Drag force represents the hydrodynamic force that exerts a body due to fluid flow passing around it (Koehl, 1996), and is represented as:
FD ¼
1 qU 2 AC D 2
ð2Þ
with q the fluid density (kg=m3 ), U the free stream velocity (m=s), A the area at which the flow is exposed to (m2 ) and C D the drag coefficient. C D is a function of Re and highly depends on the size and shape of the object, which cannot easily be deduced from experiments, but was accurately obtained from CFD. C D was obtained for the longest setae; therefore A in Eq. (2) is the area of the longest setae (b s). This provided new insight into the functional morphology of the two barnacle phenotype models. Fig. 6 shows that the drag force for both exposed and sheltered models are the same for the Re studied, even though C D was quite different. Drag force is directly proportional to the exposed surface area of the animal. Although exposed barnacles have thicker structures, the shorter length (see Table 2) minimizes the surface exposed to oncoming flow, which reduces the drag force with respect to sheltered barnacles. On the other hand, exposed barnacles have a smaller spacing ratio (i=b) resulting in lower drag force
Fig. 6. Drag force F D versus Reynolds number Re for inner and outer setae.
because of the high interference of flow from neighboring setae due to a broad viscous region (larger boundary layer thickness). The small surface area and small spacing ratio of exposed barnacles trade off of each other, resulting in similar drag force as sheltered barnacles. Fig. 6A and B show that for both exposed and sheltered models, the drag force exerted to the inner setae is slightly higher than the outer setae at the same Re. This is due to the higher viscous layer around inner setae due to the interference of the neighboring setae from all directions. 4.3. Leakiness Leakiness is defined as the ratio of the mass flow rate of fluid that flows through the gap between the inner setae to the mass flow rate of fluid at the inlet of the box, indicated with green arrows in Fig. 2. Fig. 7 shows values of leakiness for a wide range of Re from our CFD results. At around Re 1, the sheltered barnacle model demonstrated nearly three times higher leakiness through the enclosed region compared to the exposed barnacle model. The flow around exposed model at the same Reynolds, showed the characteristics of a creeping flow, indicating little to no leakiness, like a paddle (Fig. 4A). The higher velocity in between gap spaces of filters observed in CFD velocity profiles may be interpreted as fluid passing between the segments and setae, which corresponds to a leaky state (Figs. 3C and 4C). As Re increased, the leakiness of filters increases for both models, with sheltered filters showing a higher percentage passing through overall. Due to experimental restriction, we could not quantify leakiness from the laser-illumination analysis. As a best proxy, leakiness was assumed once particles were observed to pass directly through the center of a filter, as depicted with arrows in Fig. 8. At the lowest velocity tested (i.e. 0:03 m=s; Re ¼ 0:43 for sheltered and Re ¼ 0:52 for exposed), both barnacle filters were non-leaky paddles. As velocity approached 0.062 m/s (Re ¼ 0:89), comparable to our lowest CFD velocity tested, sheltered barnacles revealed slight leakiness. This velocity is higher than that measured from the sheltered collection site (Table 1). Leakiness increased as the flow velocity was increased up to 0.16 m/s (Re ¼ 2:24), after which complete leakiness was observed (Fig. 8B and Video 1). Exposed barnacles revealed onset of leakiness at 0.14 m/s (Re ¼ 2:63) with full leakiness at 0.18 m/s (Re ¼ 3:50) (Fig. 8D and Video 2). In conclusion, there is a critical range of Reynolds numbers, beyond which full leakiness is observed (sieve-like behavior) and below which no leakiness is observed (paddle-like behavior). We refer to this critical range as the leaky-paddle transition range. The leaky-paddle range for sheltered and exposed barnacles are depicted in Fig. 7 as shaded areas and lie within the results
Fig. 7. Percent of mass flow rate (leakiness) passing through inner setae at increasing Re from CFD results. The shaded areas are critical values obtained from laser illumination experiments. For Re values smaller than the critical range, leakiness was not observed (paddle-like behaviour), and for Re values larger than the critical range leakiness was observed.
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
233
Fig. 8. Laser-illumination experiments illustrating sieve-like behavior in sheltered and exposed barnacle filters with flow passing from right to left. A–C show sieve-like behavior for sheltered barnacles at U ¼ 0:16 m=s (Re ¼ 2:24). D–F show sieve-like behavior for exposed barnacles at U ¼ 0:18 m=s (Re ¼ 3:50). White arrows denote movement of a particle passing directly through the center of filter.
obtained by CFD. Our CFD and experimental results showed that leakiness for the sheltered phenotype initiates at lower Re. The transition from paddle to leaky occurs over a narrower Re range for the exposed phenotype (Fig. 7).
5. Discussion Balanus glandula has remained at the centre of marine biology investigations since Darwin (1854). The extreme modification of ramus and seta length in response to water velocity is one of the best documented ecophenotypes (Marchinko and Palmer, 2003). The majority of cirrus variation observed in nature arises from environmentally induced phenotypic plasticity (Arsenault et al., 2001; Marchinko and Palmer, 2003). The development of rami form is clearly correlated with flow velocity. Here we quantify leakiness as a function of Reynolds number (see Eq. (1)). We show that shorter, stouter feeding rami (exposed models) have the same drag force to longer and slimmer feeding rami (sheltered models) because of their smaller spacing ratio, but the range of flow velocity over which feeding appendages transition from paddle to sieve is differs between models. This comparative approach revealed details of flow in otherwise hard to see regions of B. glandula feeding rami, and revealed flow details at high flow velocities that have proven impossible to observe. This comparative approach could equally be used to compare different periods of an individuals growth, or to quantify interspecific differences at one developmental period. We chose to quantify an intraspecific phenotypic difference, because like interspecific comparisons, it is an example of developmental repatterning, but it is unique in that it sheds light on the critical marine ecological parameter, flow velocity. Another variable that affects the Reynolds number is temperature. Warmer water is less viscous. Balanus glandula experiences seasonal and latitudinal ranges in temperature. It geographic range is from Alaska to the Baja Peninsula. Temperature-induced viscosity change likely impact the feeding biology including food detection, capture, and ingestion rates (Podolsky, 1994), and it has a dramatic effects on the physiology of B. glandula (Nishizaki and Carrington, 2015).
At low Re the velocity gradient around feeding appendages produces a large viscous displacement effect, which diverts the flow away from surfaces due to the broad viscous region, consequently pushing particles away and making the filter less effective as a sieve to strain particles (Koehl and Strickier, 1981; Cheer and Koehl, 1987a). The critical range for transitioning from a paddle to a sieve for exposed Balanus glandula filters occurs between 2:63 < Re < 3:5 (0:14 < Uðm=sÞ < 0:18), while sheltered barnacles have a lower threshold of 0:89 < Re < 2:24 (0:062 < Uðm=sÞ < 0:16) for sieving. Within the critical transitioning range, the two phenotypes function very differently. The longer, slimmer geometry of sheltered barnacles provides more space for the flow to pass. Interestingly, we found that the paddle to sieve transition occurred at a higher velocity (0:062 < Uðm=sÞ < 0:16) than the maximum velocity reported from the sheltered, Grappler Narrows, collection site (0.0052 m/s, Table 1) (Marchinko and Palmer, 2003). Theoretically, the leakiness will increase feeding efficiency; but in reality, barnacles from very sheltered habitats may not experience leakiness, or must actively feed to achieve leakiness. Reasons why sheltered B. glandula do not sieve, may be ecological - a high food density in Grappler Narrows may negate the need for a sieve - or mechanical - longer legs may be more prone to breakage at the high velocities. The shorter, stout, geometry of exposed barnacles reduce drag forces, and transition to a sieve at high flow velocities, experienced in the field. The average maximum velocity in Seppings Island site is 4.41 m/s (Marchinko and Palmer, 2003), which is higher than the leaky-paddle transition for exposed barnacles obtained from our CFD results, meaning that exposed barnacles in the Seppings Island site function as sieves. The high velocity and continuous influx of particle-filled water passing through exposed filters provide a larger volume of flow per unit surface. This allows them to present less surface to the flow while maintaining the capacity to efficiently filter particles. The wake that develops downstream of barnacle feeding appendages gives information about the leakiness. This wake is controlled by Reynolds number and gap spacing (Kang, 2003). A large, slow-moving wake with counter-rotating eddies was observed from laser-illumination experiments and also in CFD simulations for both the exposed and sheltered barnacles. These wakes
234
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
are comparable to wakes formed by flow over solid bluff objects (Kang, 2003; Meneghini et al., 2001; Zhang and Zhou, 2001). At low Re, little flow passes between hairs resulting in a paddle-like function. To validate this result, a new set of simulations were performed (Fig. 9). A two-dimensional CFD simulation comparing flow through 6 uniform diameter cylinders arranged in a side-by-side arrangement normal to flow versus an array of 6 cylinders of decreasing diameters at Re ¼ 0:5 revealed similar large recirculating vortices behind segments for the uniform cylinder arrangement (Fig. 9). This indicates that uniform cylinders row is acting as a bluff object at this low Re. However, increasing the relative center to center distance to diameter ratio, allows the flow to pass through at the same Reynolds number value. Increasing the gap to diameter ratio is analogous to switching from an exposed to a sheltered barnacle. This effect evidences the ability for barnacles to adapt biological filters to the flow environment. At low Re, when cylinders are very packed, the boundary layers interfere, and acts as a wall that blocks flow between them (Kang, 2003; Meneghini et al., 2001; Zhang and Zhou, 2001). At the lower range of tested velocities, for Re lower than 1, the difference in setae spacing ratio (i=b) has little effect on the flow rate through the appendage, both behave as paddles. This is similar to what is observed in cylinder interference observations. When i=b is lower than 3, both for the exposed and sheltered barnacles (Table 2), at these low Reynolds numbers values (i.e. Re < 1), filters act as a
single bluff body keeping gap flow minimal and deflecting the flow to the outside of the appendages. As flow velocity increases, twin recirculating vortices observed with the laser-illumination experiments, for both phenotypes, indicated more gap flow rate (i.e. flow passing through the appendages) which in turn influenced wake patterns (Videos 1 and 2). The deflection of the gap flow was irregular and unsteady, resulting in a narrow and a wide vortex street. As flow separation and flow between cirri was not yet evident, gap flows were assumed to be weak, therefore the filters behave as slightly leaky paddles. As flow velocity increased further, for Re ¼ 2:24 (sheltered) and 3.5 (exposed), wakes with flow separation points located further rearward from cirri indicated that a steady flow passed through gap spaces, indicating full leakiness of the filters. Our CFD results and experimental observations are congruent with respect to general flow and wake patterns. However, limitations with the numerical models have to be stressed. All simulations were performed on passive barnacle filters, and active feeding behavior was not considered. This is a limitation because the movement of an appendage relative to fluid and towards a wall (i.e. substrate, body surface, or neighbor) can change the leakiness at low Re (Cheer and Koehl, 1987b; Koehl, 1996; Loudon et al., 1994). As barnacles are facultative suspension feeders, actively sweeping their filters in slow flow and passively extending filters held rigidly opposed to the direction of flow in faster moving water, sheltered barnacles would normally actively feed at the lower end of our velocities tested. Similarly, changes in setae spacing ratio (i=b) may affect the leakiness of a filter moving relative to a wall (Loudon et al., 1994). On the other extreme, computational simulations may overestimate drag coefficients because models are fully rigid. They do not bend and thus present the same surface to the flow for all velocities. Our laser-illumination observations, and Marchinko and Palmer (2003), Ferner and Gaylord (2008), show that sheltered barnacles undergo major cirral deformation for flow velocities above 0.21 m/s, while exposed cirral fans undergo minor cirral deformation at 0.49 m/s. The two simulation models represent only 6 medially located rami segments (Fig. 9), to keep computational time and calculations reasonable. This enabled us to analyze both inner, interdigitated setae, and outer setae that are open to flow. A complete cirral filter under a range of Re may be expected to provide more realistic estimates on the transition from paddle to sieve. Another limitation with the numerical models was that the feeding appendages in the model were assumed smooth elliptical cylinders, while, arthropod exoskeletons are composite materials that have complex structural reactions to flow with surfaces that are typically not smooth. Despite these limitations, these CFD simulations, validated against direct observations, show that Balanus glandula from exposed sites are remarkably adaptable to a wide range of flow velocities, whereas those from sheltered sites may not be mechanically optimal. Their abundance in sheltered bays however, suggest there whatever the reason, ecological, developmental, or mechanical, the trade-off is a good one. The beauty of biological fluid mechanics, is that the physical selection pressure can be precisely quantified, and the fluid forces have not changed throughout the origin and evolution of life. Conflict of interest statement The authors have nothing to disclose. Acknowledgments
Fig. 9. Velocity contours through (A) uniform cylinders and (B) cylinders of decreasing size.
Thank you to E. Tixier, R. Yu, and S. Corbeil-Letourneau at Ecole Polytechnique for discussions and assistance with finite volume
M. Vo et al. / Journal of Biomechanics 71 (2018) 225–235
models, and CFD, and to the director and staff of the Bamfield Marine Sciences Centre for logistic support. Conversations with A.R. Palmer (University of Alberta) were central to our decision to embark on this project. We are grateful to Jason Hodin, Mike Nishizaki and one anonymous reviewer for improvements to this publication. This project was funded by an NSERC Discovery grant to C.B. C. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.jbiomech.2018.02. 011. References Arsenault, D.J., Marchinko, K.B., Palmer, A.R., 2001. Precise tuning of barnacle leg length to coastal wave action. Proc. R. Soc. London, Ser. B 268 (1481), 2149– 2154. Cheer, A.Y.L., Koehl, M.A.R., 1987a. Fluid flow through filtering appendages of insects. Math. Med. Biol. 4 (3), 185–199. Cheer, A.Y.L., Koehl, M.A.R., 1987b. Paddles and rakes: fluid flow through bristled appendages of small organisms. J. Theor. Biol. 129 (1), 17–39. Darwin, C., 1854. A Monograph on the Sub-Class Cirripedia, With Figures of all the Species, vol. 1. Ray society, London. URL
235
Koehl, M.A.R., Strickier, J.R., 1981. Copepod feeding currents: food capture at low Reynolds number. Limnol. Oceanogr. 26 (6), 1062–1073. LaBarbera, M., 1984. Feeding currents and particle capture mechanisms in suspension feeding animals. Am. Zool. 24 (1), 71–84. Larsson, A.I., Jonsson, P.R., 2006. Barnacle larvae actively select flow environments supporting post-settlement growth and survival. Ecology 87 (8), 1960–1966. Loudon, C., Best, B., Koehl, M.A.R., 1994. When does motion relative to neighboring surfaces alter the flow through arrays of hairs? J. Exp. Biol. 193 (1), 233–254. Marchinko, K.B., 2007. Feeding behavior reveals the adaptive nature of plasticity in barnacle feeding limbs. Biol. Bull. 213 (1), 12–15. Marchinko, K.B., Geary, D., 2003. Dramatic phenotypic plasticity in barnacle legs (Balanus glandula Darwin): magnitude, age dependence, and speed of response. Evolution 57 (6), 1281–1290. Marchinko, K.B., Palmer, A.R., 2003. Feeding in flow extremes: dependence of cirrus form on wave-exposure in four barnacle species. Zoology 106 (2), 127–141. Mehrabian, S., Letendre, F., Cameron, C.B., 2018. The mechanisms of filter feeding on oil droplets: theoretical considerations. Mar. Environ. Res., doi:https://doi.org/ 10.1016/j.marenvres.2018.01.006 (in press). Meneghini, J., Saltara, F., Siqueira, C., Ferrari, J., 2001. Numerical simulation of flow interference between two circular cylinders in tandem and side-by-side arrangements. J. Fluids Struct. 15 (2), 327–350. Miller, L.P., 2007. Feeding in extreme flows: behavior compensates for mechanical constraints in barnacle cirri. Mar. Ecol. Prog. Ser. 227. Nishizaki, M.T., Carrington, E., 2015. The effect of water temperature and velocity on barnacle growth: quantifying the impact of multiple environmental stressors. J. Therm. Biol. 54, 37–46. Podolsky, R.D. et al., 1994. Temperature and water viscosity: physiological versus mechanical effects on suspension feeding. Science 265, 100–103. Riisgård, H.U., Larsen, P.S., 2010. Particle capture mechanisms in suspensionfeeding invertebrates. Mar. Ecol. Prog. Ser. 418, 255–293. Rubenstein, D.I., Koehl, M.A.R., 1977. The mechanisms of filter feeding: some theoretical considerations. Am. Nat. 111 (981), 981–994. Schneider, C.A., Rasband, W.S., Eliceiri, K.W., 2012. NIH image to ImageJ: 25 years of image analysis. Nat. Meth. 9 (7), 671. Shimeta, J., Jumars, P.A., 1991. Physical mechanisms and rates of particle capture by suspension-feeders. Oceanogr. Mar. Biol. Annu. Rev 29 (19), l-257. Southward, A., 1955. On the behaviour of barnacles ii. the influence of habitat and tide-level on cirral activity. J. Mar. Biol. Assoc. U.K. 34 (03), 423–433. Tamada, K., Fujikawa, H., 1957. The steady two-dimensional flow of viscous fluid at low Reynolds numbers passing through an infinite row of equal parallel circular cylinders. Trager, G.C., Hwang, J.S., Strickler, J.R., 1990. Barnacle suspension-feeding in variable flow. Mar. Biol. 105 (1), 117–127. Vogel, S., 2003. Comparative Biomechanics: Life’s Physical World. Princeton, New Jersey, USA. Vogel, S., Savage, A.A., 1996. Life in moving fluids. The physical biology of flow. Biol. J. Linn. Soc. 57 (3), 291. Zhang, H.J., Zhou, Y., 2001. Effect of unequal cylinder spacing on vortex streets behind three side-by-side cylinders. Phys. Fluids 13 (12), 3675–3686.